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Journal of the Physical Society of Japan

Critical Temperature, Critical Current, and Vortex Loops in 3D Anisotropic Josephson Junction Arrays

Chikao Kawabata, Masaki Takeuchi, Subodh R. Shenoy, A. R. Bishop

Published Jan 1, 2000• 5 citations
Publisher

Abstract

Monte Carlo simulations of a 3D anisotropic Josephson junction array are performed, to determine the critical temperature T c ( I ,α) for the onset of phase incoherence. Here I is the external current drive at opposite surfaces, and α≡ E J ⊥ / E J // is the inter-/intra-layer Josephson coupling ratio. T c ( I ,α) is equivalent to a critical current, I c ( T ,α) for the breakdown of global phase coherence. The temperature and anisotropy dependence of I c ( T ,α) supports a physical picture of two vortex populations, quasi-2D and 3D in nature at small and large scales respectively, separated by a length scale \(r_{\perp}\sim 1/\sqrt{\alpha}\).

DOI
10.1143/jpsj.69.194

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Monte Carlo simulations of a 3D anisotropic Josephson junction array are performed, to determine the critical temperature T c ( I ,α) for the onset of phase incoherence. Here I is the external current drive at opposite surfaces, and α≡ E J ⊥ / E J // is the inter-/intra-layer Josephson coupling ratio. T c ( I ,α) is equivalent to a critical current, I c ( T ,α) for the breakdown of global phase coherence. The temperature and anisotropy dependence of I c ( T ,α) supports a physical picture of two vortex populations, quasi-2D and 3D in nature at small and large scales respectively, separated by a length scale (r_{\perp}\sim 1/\sqrt{\alpha}).
Motivated by the interplay between 2D and 3D scaling signatures observed in unconventional layered superconductors, we present a systematic Monte Carlo study of the three-dimensional classical XY model with anisotropic in-plane J∥J_\parallelJ∥​ and inter-plane J⊥J_\perpJ⊥​ couplings. Our study includes very small values of the system anisotropy Δ=J⊥/J∥Δ=J_\perp /J_\parallelΔ=J⊥​/J∥​ not studied before, and focuses on characterizing the crossover from quasi-2D topological scaling to genuine 3D critical behavior. The numerical results for the critical temperature unambiguously reveal a logarithmic scaling with ΔΔΔ, directly related to the topological scaling in the 2D limit. Despite the 3D nature of the layered XY criticality, topological scaling signatures survive up to system sizes comparable to the crossover length ℓJ\ell_JℓJ​, which diverges at small ΔΔΔ with a scaling behavior reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) transition. This shows that genuine 3D symmetry-breaking behavior emerges only at exceedingly large system sizes when the anisotropy is very strong. Our results indicate that new experimental evidence is required to clarify the extent to which the critical signatures observed in layered strongly correlated materials are shaped by their pronounced anisotropy.
Scaling equations are obtained for thermally excited vortex loops in the anisotropic three-dimensional (3D) XY model with interplane/intraplane coupling ratio ${\mathit{K}}_{\mathrm{\ensuremath{\perp}}}$/${\mathit{K}}_{\mathrm{\ensuremath{\parallel}}}$=${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}2}$. For high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors, ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}2}$, related to the Ginzburg-Landau masses, is in a strong anisotropy regime, ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$0.5, and a length scale r0{\mathit{r}}_{0}r0​=${\ensuremath{\gamma}}_{0}$a0{\mathit{a}}_{0}a0​ naturally arises (a0{\mathit{a}}_{0}a0​= lattice constant). (i) For loop major axes a>r0{\mathit{r}}_{0}r0​, the dominant excitations are 3D elliptical vortex loops cutting multiple planes, with a\ensuremath{\sim}${\ensuremath{\xi}}_{\mathrm{\ensuremath{-}}}$(T)=a0{\mathit{a}}_{0}a0​\ensuremath{\Vert}\ensuremath{\varepsilon}${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ where \ensuremath{\Vert}\ensuremath{\varepsilon}\ensuremath{\Vert}\ensuremath{\equiv}\ensuremath{\Vert}(T-Tc{\mathit{T}}_{\mathit{c}}Tc​)/Tc{\mathit{T}}_{\mathit{c}}Tc​\ensuremath{\Vert}, and \ensuremath{\nu}=0.67. The vorticity segment components (\ensuremath{\mu}=x^,y^,z^) are ${\mathit{J}}_{\mathrm{\ensuremath{\mu}}}$(r)=0,\ifmmode\pm\else\textpm\fi{}1, and interact via a Biot-Savart-like law. The renormalized anisotropy ${\ensuremath{\gamma}}_{\mathit{l}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}1, asymptotically isotropic, as l=ln(a/a0{\mathit{a}}_{0}a0​)\ensuremath{\rightarrow}\ensuremath{\infty}: anisotropy is irrelevant. (ii) For loop scales r0{\mathit{r}}_{0}r0​>a>2a0{\mathit{a}}_{0}a0​ the dominant excitations are quasi-2D rectangular loops with short sides Jz{\mathit{J}}_{\mathit{z}}Jz​(r)=\ifmmode\pm\else\textpm\fi{}1 cutting single planes, that are thus effectively decoupled at finite scales r0{\mathit{r}}_{0}r0​. The Jz{\mathit{J}}_{\mathit{z}}Jz​=\ifmmode\pm\else\textpm\fi{}1 vortex components, of in-plane separation \ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{W}}$mN\ensuremath{\Vert}r0{\mathit{r}}_{0}r0​, interact via a logarithmic [ln(\ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\Vert}/r0{\mathit{r}}_{0}r0​)] plus linear [Q0{\mathit{Q}}_{0}Q0​(\ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\Vert}/a0{\mathit{a}}_{0}a0​-1)] potential. As ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}0, the coefficient Q0{\mathit{Q}}_{0}Q0​\ensuremath{\rightarrow}0, and the Kosterlitz-Thouless limit is recovered. (iii)The 3D transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ versus ${\mathit{K}}_{\mathrm{\ensuremath{\perp}}}$/${\mathit{K}}_{\mathrm{\ensuremath{\parallel}}}$, calculated for strong anisotropies from ``2D'' (3D) scaling for scales r0{\mathit{r}}_{0}r0​ (>r0{\mathit{r}}_{0}r0​), matches existing Monte Carlo data well. Critical regions \ensuremath{\Vert}${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$\ensuremath{\Vert}\ensuremath{\le}${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1/\ensuremath{\nu}}$ are estimated. Contact is made with ideas of an intrinsic critical current arising from linear \ensuremath{\sim}${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$${\mathit{R}}_{\mathrm{\ensuremath{\parallel}}}$ vortex segment effective potentials.
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Superconductor-semiconductor Josephson junction arrays are a uniquely tunable platform for studying collective quantum phenomena, particularly in the regime where localized Andreev bound states can hybridize across the lattice when the physical separation between adjacent junctions is smaller than their coherence length (ξABS>dJJξ_{\text{ABS}}>d_{\text{JJ}}ξABS​>dJJ​). Here, we investigate three distinct Al-InAs Josephson junction arrays: a square array and super-honeycomb array fabricated within this ξABS>dJJ{ξ_{\text{ABS}}>d_{\text{JJ}}}ξABS​>dJJ​ regime, as well as a larger-spacing super-honeycomb control device designed such that ξABS≲ ⁣ dJJξ_{\text{ABS}}\lesssim\!~d_{\text{JJ}}ξABS​≲ dJJ​. Under an out-of-plane field, critical current peaks emerge at rational filling factors, reflecting stable vortex configurations in the lattices. In the super-honeycomb lattice, vortices localize to distinct non-identical plaquettes at different filling factors, as predicted by frustrated XY model simulations. A rotating in-plane field yields periodic critical current oscillations that reflect the Rashba spin-orbit coupling inherent to the InAs quantum well. Surprisingly, at f=1f = 1f=1, the closely spaced super-honeycomb array exhibits a distinct critical current minimum as the magnitude of the in-plane field increases, a signature absent in the square array and large-spacing super-honeycomb array. These results indicate that this signature is jointly influenced by the unique geometry of the super-honeycomb vortex lattice and by long-range inter-junction hybridization.
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We study the differential resistivity transition of two-dimensional superconducting arrays induced by an external driving current, in the presence of thermal fluctuations and a magnetic field corresponding to fff flux quantum per plaquette. Recent experiments have identified this transition as a dynamic vortex Mott insulator transition at vortex densities near rational values of fff. The critical behavior is determined from a scaling analysis of the current-voltage relation near the transition, obtained by Monte Carlo simulations of a Josephson-junction array model in the vortex representation. For a square-lattice array, the critical exponents obtained near f=1/2f=1/2f=1/2 are consistent with the experimental observations. The same scaling behavior is observed near f=1/3f=1/3f=1/3. For a honeycomb array, although similar results are obtained for f=1/3f=1/3f=1/3, the transition is absent for f=1/2f=1/2f=1/2, consistent with an incommensurate vortex phase.